Optimal policy for inventory transfer between two depots with backlogging

This technical note considers the optimal control problem of transferring empty containers between two depots over a multiperiod planning horizon to minimize the total cost comprising inventory holding costs, empty container transfer costs, and demand backlog costs. The problem involves random supply and random demand. Formulating the problem as a stochastic dynamic program, we show that the value function is not convex so the traditional method of analysis cannot be applied. We present an alternative approach by focusing on the local properties of the value function such as the first and second derivatives on a region-wise basis. This enables us to establish the structural characteristics of the optimal policy, e.g., several monotonic switching curves divide the state space into seven control regions. Based on the established structural properties, we develop a simple near-optimal policy. We provide a numerical example to illustrate the analytical results.